# Phenomenology of \(U(1)_{F}\) extension of inert-doublet model with exotic scalars and leptons

## Abstract

In this work we will extend the inert-doublet model (IDM) by adding a new \(U(1)_{F}\) gauge symmetry to it, under which, a \(Z_{2}\) even scalar (\(\phi _{2}\)) and \(Z_{2}\) odd right handed component of two exotic charged leptons \((F_{eR},\ F_{\mu R})\), are charged. We also add one \(Z_{2}\) even real scalar (\(\phi _{1}\)) and one complex scalar (\(\phi \)), three neutral Majorana right handed fermions (\(N_{1},\ N_{2},\ N_{3}\)), two left handed components of the exotic charged leptons \((F_{eL},\ F_{\mu L})\) as well as \(F_{\tau }\) are all odd under the \(Z_{2}\), all of which are not charged under the \(U(1)_{F}\). With these new particles added to the IDM, we have a model which can give two scalar DM candidates, together they can explain the present DM relic density as well as the muon (g-2) anomaly simultaneously. Also in this model the neutrino masses are generated at one loop level. One of the most peculiar feature of this model is that non-trivial solution to the axial gauge anomaly free conditions lead to the prediction of a stable very heavy partner to the electron (\(F_{e}\)), whose present collider limit (13 TeV LHC) on its mass should be around \(m_{F_{e}} \ge \) few TeV.

## 1 Introduction

The self consistency of the standard-model (SM) has been established (tentatively at least) with LHC discovery of the scalar behaving like the Higgs scalar of the SM. And about years ago since the SM has been formulated, it has went through numerous experimental test and no major disagreement with its prediction has been discovered. But still our universe turn out to be at least little more complicated than the SM can anticipate. One major discovery that point towards incompleteness of SM is the presence of dark-matter (DM). Another was the discovery of neutrino oscillation which means that neutrinos has small masses where as in SM neutrinos are massless. Then also it turn out that CP violation in SM due to Kobayashi-Maskawa (KM) [1] phase turn out to be too small to explain the observed excess of matter over the anti-matter. So the most important questions that theoretical and experimental efforts in years to come will be related to nature of DM, mechanism behind the neutrino oscillation phenomena and search for new sources of CP violations.

In this work we will extend the SM by adding one more \(U(1)_{F}\) guage group to it with introducing only new exotic scalars and leptons. This simple extension turn out to explain the DM relic density, loop generated neutrino masses, Baryon genesis and muon (g-2) anomaly. One of the peculiar features of this particular realization of \(U(1)_{F}\) extension of SM is that, due to solutions to the axial anomaly free conditions, if the electric charge of \(F_{\tau }\) is taken as a free parameter, then the electric charge of \(F_{e}\) will have opposite sign to that of the \(F_{\mu }\), so if \(F_{\mu }\) to explain the observed deviation in muon (g-2) then \(F_{e}\) is require to be stable.

This paper is organized as follows. In the next section we introduce various particles in our model and write down Lagrangian invariant under all the symmetries imposed on the particles. Then in section III and IV we dwell into the consequences of the model related to muon (g-2), neutrino mass, DM, Baryon-Genesis and collider signature of the exotic charged leptons. We conclude in section V.

## 2 Model details

The charge assignments of new particles under the full gauge group \(SU(3)_{c}\times SU(2)_{L}\times U(1)_{Y}\times U(1)_{F}\). Our choice of \(U(1)_{F}\) charge of \(\phi _{2}\) is related to our choice of particular solution of the axial anomaly free equations. In this particular choice, \(\phi _{2}\) gives masses to \(F_{e}\) and \(F_{\mu }\) and \(\phi _{1}\) gives mass to \(F_{\tau }\). Where index \(i = e, \mu , \tau \) and our Y of SM \(U(1)_{Y}\) is same as \(\frac{Y}{2}\) in the usual convention

Particles | \(SU(3)_{c}\) | \(SU(2)_{L}\) | \(U(1)_{Y}\) | \(U(1)_{F}\) | \(Z_{2}\) |
---|---|---|---|---|---|

\(F_{iR}\) | 1 | 1 | \(Y_{i}\) | \(n_{i}\) | \(-1\) |

\(F_{iL}\) | 1 | 1 | \(Y_{i}\) | 0 | \(-1\) |

\(N_{iR}\) | 1 | 1 | 0 | 0 | \(-1\) |

\(\phi \) | 1 | 1 | 0 | 0 | \(-1\) |

\(\phi _{1}\) | 1 | 1 | 0 | 0 | \(+1\) |

\(\phi _{2}\) | 1 | 1 | 0 | \(n_{\phi _{2}} = n_{\mu } = -n_{e}\) | \(+1\) |

\(\eta \) | 1 | 2 | 1/2 | 0 | \(-1\) |

A non-trivial solution of Eqs. (1) and (2) for \(Y_{i}\)s and \(n_{i}\)s that is interesting is given when we set either \(n_{e}\) or \(n_{\mu }\) or \(n_{\tau }\) equal to zero. In this work we set \(n_{\tau } = 0\) which make the electric charge of \(F_{\tau }\) a free parameter. Then the four equations are solved for \(Y_{e}\), \(Y_{\mu }\), \(n_{e}\) and \(n_{\mu }\) if we set \(Y_{e} = -Y_{\mu }\) and \(n_{e} = -n_{\mu }\), but for \(F_{\mu }\) to explain the observed anomaly in \(\delta a_{\mu }\) [3], we require \(Y_{\mu } = Q_{F_{\mu }} = -1\) which implies that \(Y_{e} = Q_{F_{e}} = +1\) and so charge conservation and Lorenz invariance forbid \(F_{e}\) to contribute to \(\delta a_{e}\), which is consistent with the experimental findings as no deviation in \(\delta a_{e}\) from the SM prediction has been reported. If we require Yukawa terms such as \(y_{\mu }\bar{\mu }_{R}F_{\mu L}\phi \), to explain the observed anomaly in muon (g-2), then term such \(y_{e}\bar{e}_{R}F_{e L}\phi \) are forbidden by charge conservation and term such as \(y_{e}\bar{e}_{R}F_{e L}^{c}\phi \) is forbiden by Lorenz invariance, so \(F_{e}\) will be a stable heavy charged particle. Although \(Y_{\tau }\) is a free parameter, if we also set \(Y_{\tau } = Q_{F_{\tau }} = -1\), then we can expect a deviation in \(\delta a_{\tau }\) in future measurements coming from \(y_{\tau }\bar{\tau }_{R}\phi F_{\tau }\) term.

### 2.1 Scalar sector

### 2.2 Fermionic sector

## 3 Muon (g-2) anomaly

## 4 Loop generation of neutrino masses and dark-matter

## 5 Conclusions

In this work we have proposed a simple extension of the SM by only adding three \(Z_{2}\) odd exotic charged leptons (\(F_{e},\ F_{\mu },\ F_{\tau }\)) whose right handed component are charged under a new \(U(1)_{F}\) gauge symmetry, three \(Z_{2}\) odd neutral fermions (\(N_{1},\ N_{2},\ N_{3}\)) singlet under the entire gauge symmetry, one \(SU(2)_{L}\) doublet (\(\eta \)) odd under a discrete \(Z_{2}\) symmetry whose VEV is zero (the inert-doublet model), one \(Z_{2}\) odd scalar (\(\phi \)) singlet under the entire gauge group and whose VEV is zero, one \(Z_{2}\) even scalar (\(\phi _{1}\)) also singlet under the entire gauge group which develops a non zero VEV \(v_{1}\), one more \(Z_{2}\) even scalar (\(\phi _{2}\)) charged under the \(U(1)_{F}\) whose non-zero VEV breaks the \(U(1)_{F}\) gauge symmetry spontaneously and give mass to the gauge boson \(Z_{F}\). With addition of the above new particles, we have been able to build a model which can give two DM candidate in terms of lightest neutral component of the inert-doublet (\(\eta _{R}\) in our case) and \(\phi \) but if \(\phi \) to explain the muon (g-2) anomaly than \(\phi \) will contribute only a tiny fraction to the present DM relic density, so most of the present relic density of DM will consist of \(\eta _{R}\) whose mass can be in the range 500 GeV to 130 TeV. In this model neutrino masses are generated at one loop level as well as baryon-genesis via lepto-genesis is also possible. We have also given key signatures of these new exotic leptons at LHC and future \(e^{+}e^{-}\) colliders. We find the the key signature will be in the form of \(e^{+}e^{-}/pp \rightarrow Z^{*}\gamma ^{*} \rightarrow F_{\mu }^{+}F^{-}_{\mu }(F_{\tau }^{+}F^{-}_{\tau }) \rightarrow \mu ^{+}\mu ^{-}(\tau ^{+}\tau ^{-}) + missing\ energy\ (\phi \phi )\) and \(e^{+}e^{-}/pp \rightarrow Z^{*}\gamma ^{*} \rightarrow \gamma + \phi _{2} (via\ triangle\ loop) \rightarrow \gamma + missing\ energy\ (\phi _{2} \rightarrow \phi \phi )\). One of the most peculiar signature of this model is the existence of a stable very heavy (about \(m_{F_{e}} \ge \) few TeV) charged lepton partner to the electron.

## Notes

### Acknowledgements

This work is supported and funded by the Department of Atomic Energy of the Government of India and by the Government of U.P.

## References

- 1.M. Kobayashi, T. Maskawa, Prog. Theor. Phys.
**49**, 652 (1973)ADSCrossRefGoogle Scholar - 2.E. Ma, Phys. Rev. D
**73**, 077301 (2006)ADSCrossRefGoogle Scholar - 3.L. Dhargyal, (2017). arXiv:1705.09610v2 [hep-ph]
- 4.T. Yanagita, (KEK Report No. 79-18, Tsukuba, Japan, 1979), p. 95Google Scholar
- 5.M. Gell-Mann, P. Ramond, R. Slansky, in
*Supergravity*, ed. by P. von Nieuwenhuizen, D.Z. Freedman (Amsterdam, 1979), p. 315Google Scholar - 6.R. Mohapatra, G. Senjanovic, Phys. Rev. Lett.
**44**, 912 (1980)ADSCrossRefGoogle Scholar - 7.M. Fukugita, T. Yanagita, Phys. Lett. B
**174**, 45 (1986)ADSCrossRefGoogle Scholar - 8.J.P. Leveille, Nucl. Phys. B
**137**, 63–76 (1978)ADSCrossRefGoogle Scholar - 9.T. Abe, R. Sato, K. Yagyu, (2017) arXiv:1705.01469 [hep-ph]. https://doi.org/10.1007/JHEP07(2017)012
- 10.C.W. Chaing, H. Okada, E. Senaha, Phys. Rev. D
**96**(1), 015002 (2017). arXiv:1703.09153v2 [hep-ph]ADSCrossRefGoogle Scholar - 11.C. Patrignani et al. (Particle Data Group), Chin. Phys. C, 40, 100001 (2016) and 2017 updateGoogle Scholar
- 12.E. Ma, (2006). arXiv:hep-ph/0605180v2
- 13.W. Buchmuller, R.D. Peccei, T. Yanagita, Ann. Rev. Nucl. Part. Sci.
**55**, 311 (2005)ADSCrossRefGoogle Scholar - 14.T. Hambey, F.S. Ling, L.L. Honorez, J. Rocher, JHEP
**07**, 090 (2009)ADSCrossRefGoogle Scholar - 15.T. Hambey, F.S. Ling, L.L. Honorez, J. Rocher, JHEP
**05**, 66 (2010)ADSCrossRefGoogle Scholar - 16.L.L. Honorez, E. Nezri, J.F. Oliver, M.H.G. Tytgat, JCAP
**02**, 028 (2007)CrossRefGoogle Scholar - 17.A. Goudelis, (2015). arXiv:1510.03993v1 [hep-ph]
- 18.A. Goudelis, B. Herrmann, O. St\(\hat{a}\)l, https://doi.org/10.1007/JHEP09(2013)106
- 19.D.S. Akerib et al. (LUX Collaboration), (2017). arXiv:1608.07648v3 [astro-ph.CO]
- 20.A. Tan et al., PandaX-II Collaboration. Phys. Rev. Lett.
**117**, 121303 (2016)ADSCrossRefGoogle Scholar - 21.K. Griest, M. Kamionkowski, Phys. Rev. Lett.
**64**, 615 (1990)ADSCrossRefGoogle Scholar

## Copyright information

**Open Access**This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Funded by SCOAP^{3}